Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}5x-6y &= 4 \\ -x-3y &= 9\end{align*}$
Solution: Begin by moving the $y$ -term in the second equation to the right side of the equation. $-x = 3y+9$ Divide both sides by $-1$ to isolate $x$ $x = {-3y - 9}$ Substitute this expression for $x$ in the first equation. $5({-3y - 9}) - 6y = 4$ $-15y - 45 - 6y = 4$ Simplify by combining terms, then solve for $y$ $-21y - 45 = 4$ $-21y = 49$ $y = -\dfrac{7}{3}$ Substitute $-\dfrac{7}{3}$ for $y$ in the top equation. $5x-6( -\dfrac{7}{3}) = 4$ $5x+14 = 4$ $5x = -10$ $x = -2$ The solution is $\enspace x = -2, \enspace y = -\dfrac{7}{3}$.